三维晶体学群的极大有限群的代数基础

ALGEBRAIC FOUNDATION OF MAXIMAL FINITE GROUP OF 3 DIMENTIONAL CRYATALLOGRAPHY

从 NTN =T这一晶体学的普遍公式出发 ,推导出在三维空间中度量张量矩阵T有四个算术不等价类 ,即T1 =10 00 100 0 1,T2 =2 1012 00 0 2,T3=2 1112 1112,T4=3- 1- 1- 13- 1- 1- 13,而T1 ,T3,T4属几何等价类 ,故几何不等价类只有T1 及T2 .根据 NT1 N =T1 及 NT2 N =T2 求出三维晶体学的两个极大有限群分别为 48阶及 2 4阶 ,它们对应于两个晶体学点群 。

Four arithmetic non-equivalent metric tenser matrices, T-1 = (0(0)(1) 1(0)(0) 0(1)(0)), T-2 = (1(0)(2) 2(0)(1) 0(2)(0)), T-3 = (1(1)(2) 2(1)(1) 1(2)(1)) and T-4 = (-1(-1)(3) 3(-1)(-1) -1(3)(-1)) have been derived in this paper according to a crystallographic general equation (N) over tilde TN> = T. T-1, T-3 and T-4 are geometric equivalent ones, therefore, only T-1 and T-2 are geometric non-equivalent ones.Substituting T-1 and T-2 into (N) over tilde TN = T, two maximal finite groups can be derived, which have 48 and 24 elements respectively and belong to two crystallographic point groups. The other 30 point groups can be derived according to group-subgroup relationship.